The angle between the line x = y = z and the plane x + y + z = 4 is 90⁰.

 

$\frac{x-0}{1}=\frac{y-0}{1}=\frac{z-0}{1}$ 

x + y + z = 4

$\theta =\frac{\pi }{2}$

${\hat{x}}_1=\overline{OA}\times \overline{OB}=\left|\begin{array}{ccc}i& j& k\\ 1& 1& 1\\ 2& 1& 3\end{array}\right|$

${\hat{x}}_2=\overline{AB}\times \overline{AC}=\left|\begin{array}{ccc}i& j& k\\ 1& -1& 2\\ -2& -1& 1\end{array}\right|$

$\cos \theta =\frac{5+5+9}{\sqrt{25+1+9}\sqrt{25+1+9}}=\frac{19}{35}$

$\theta ={\cos }^{-1}\left(\frac{19}{35}\right)$

Both I and II correct